# Let us import python package Numpy (numeric python) - for carrying out the numeric calculations
import numpy as np

# And also import the python package - Matplotlib.pyplot to plot graphs
import matplotlib.pyplot as plt


#Let us define a linear function called line0 = 2x + 1
def line0(x):
    y = 2*x+1
    return y


#Matplotlib documentation: https://matplotlib.org/stable/index.html

#Define a function 'graph' to plot a chart
def graph(x,f, title):
    '''
    This function is used to plot a line chart using matplotlib.pyplot method given the below inputs
    x: Python 1d array or a list representing the value of independent variable (x)
    f: Python function with an independent input variable (x)
    Returns: Nothing
    '''
    
    # First of all, let us find the value of output of a given function with given inputs
    y = f(x)
    
    # Now, let us print the value of independent(x) & dependent(y) variables
    print('\nx:',x)
    print('\ny:',y)

    # Add x-axis and y-axis lines
    plt.axhline(y=0, color='green', linewidth=1)
    plt.axvline(x=0, color='green', linewidth=1)

    # Plot the graph
    plt.plot(x, y, label=title,linewidth=2)

    # Add labels and title
    plt.xlabel("x")
    plt.ylabel("y")
    plt.title("Graph of "+title)

    # Add a grid for better readability
    plt.grid(True)

    # Add a legend
    plt.legend()
    
    # Now, let us display the graph
    plt.show()

    return

# Now, let us create X array with evenly spaced values using np.linspace
# Ref: https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy-linspace

x = np.linspace(-10, 10, 21) # Generate range of 21 values from -10 to 10


#Now, let us see the plot using the 'graph' function
graph(x,line0,"y = 2x+1")

x: [-10.  -9.  -8.  -7.  -6.  -5.  -4.  -3.  -2.  -1.   0.   1.   2.   3.
   4.   5.   6.   7.   8.   9.  10.]

y: [-19. -17. -15. -13. -11.  -9.  -7.  -5.  -3.  -1.   1.   3.   5.   7.
   9.  11.  13.  15.  17.  19.  21.]

#define another function we want to plot, now using Python's 'lambda' method
#Refer lambda function documentation: https://realpython.com/python-lambda/

line1 = lambda x: 5*x - 6               # y = 5x-6

#Now, let us see the plot using the 'graph' function
graph(x,line1,"y = 5x-6")

x: [-10.  -9.  -8.  -7.  -6.  -5.  -4.  -3.  -2.  -1.   0.   1.   2.   3.
   4.   5.   6.   7.   8.   9.  10.]

y: [-56. -51. -46. -41. -36. -31. -26. -21. -16. -11.  -6.  -1.   4.   9.
  14.  19.  24.  29.  34.  39.  44.]

# Let us define more complex function
parabola1 = lambda x: x**2 - 25 


#Now, let us plot it as well
graph(x,parabola1,"y=x^2 - 25")

x: [-10.  -9.  -8.  -7.  -6.  -5.  -4.  -3.  -2.  -1.   0.   1.   2.   3.
   4.   5.   6.   7.   8.   9.  10.]

y: [ 75.  56.  39.  24.  11.   0.  -9. -16. -21. -24. -25. -24. -21. -16.
  -9.   0.  11.  24.  39.  56.  75.]

# Now, let us import more powerful plotting library package called plotly.express
import plotly.express as px 

#Ref:https://www.geeksforgeeks.org/python-plotly-tutorial/

#find the value of parabola function for given range of x
y = parabola1(x)

# Plot the graph of the same x & z calculated above, using plotly
fig = px.line(x=x, y=y, title='Plotly chart for equation: y=x^2 - 25') 
fig.show()

# Let us define the two functions for which we want to solve the problem using lambda method
line1 = lambda x: 2*x-1
line2 = lambda x: x+1

# Defining the input (independent) variable range
x = np.linspace(-5, 5, 21)

# Plot the graph
fig = px.line(x=x, y=[line1(x), line2(x)], title='Solving linear equation with two variables with graphing')
fig.write_html('graphing.html')
fig.show()

# Let us define the two functions for which we want to solve the problem using lambda method
para1 = lambda x: (x**2 - 25)
line2 = lambda x: (x + 5)

# Defining the input variable array
x = np.linspace(-10, 10, 101)

# Plot the graph
fig = px.line(x=x, y=[para1(x), line2(x)]) 
fig.show()

# Define function to solve linear equation by using substitution method:
def solve_by_substitution1(m1,c1,m2,c2):
    '''
    This function solves two linear equations (expressed as y = mx + c) using the substitution method
    Input m1: Slope of equation 1
          c1: Intersect of equation 1
          m2: Slope of equation 2
          c2: Intersect of equation 1
    Returns : The values of x and y
    '''
    x = - (c1-c2)/(m1-m2)
    y = m2*x + c2
    return x,y


#Call the function to solve the linear equation system with substitution system
solve_by_substitution1(-1,3, 1, -1)

(2.0, 1.0)

# Define function to solve linear equation by using substitution method:
def solve_by_substitution2(a1,b1,d1,a2,b2,d2):
    '''
    If the equation is in the form: ax +  b = d 
    
    Then we use this wrapper function over the above function 
     by replacing m1 by -a1/b1, c1 by d1/b1  and likewise m2 and c2
    
    Inputs:
      a1,b1,d1: Parameters for linear equation 1
      a2,b2,d2: Parameters for linear equation 2
    Returns: The values of x and y
    '''
    
    return solve_by_substitution1(-a1/b1, d1/b1, -a2/b2, d2/b2)


#Call the function to solve the linear equation system with substitution system
solve_by_substitution2(1,1,3,1,-1,1)

(2.0, 1.0)

# Let us define a function by using an echelon method
def solve_by_echelon(a1,b1,d1,a2,b2,d2):
    '''
    This solves the linear equation by the echelon method as per steps defined below:

    - Equation 1 - Multiply by a2:  a1x + b1y = d1,   => a1a2x + a2b1y = a2d1
    - Equation 2 - Multiply by –a1: a2x + b2y = d2  => -a1a2x – a1b2y = -a1d2
    - Add both to cancel x term  => a2b1y - a1b2y = (a2d1 - a1d2)
    - Solve for y 			  	 => y = (a2d1 - a1d2) / (a2b1 - a1d2)
    - Substitute y and solve for x     => x = (d1 – b1y)/a1
    
    Inputs:
      a1,b1,d1: Parameters for linear equation 1
      a2,b2,d2: Parameters for linear equation 2
    Returns: The values of x and y    
    '''
    y = (a2*d1 - a1*d2)/(a2*b1 - a1*b2)
    x = (d1 - b1*y)/a1
    return x,y

#Call the function to solve the linear equation system with substitution system
solve_by_echelon(1,1,3,1,-1,1)

(2.0, 1.0)

#Ref: https://www.geeksforgeeks.org/python-solve-the-linear-equation-of-multiple-variable/

# importing library sympy - Python library for symbolic mathematics
from sympy import symbols, Eq, solve 
  
# defining symbols used in equations 
# or unknown variables 
x, y = symbols('x,y') 
  
# defining equations 
eq1 = Eq((x+y), 3) 
print("\nEquation 1:", eq1) 

eq2 = Eq((x-y), 1) 
print("\nEquation 2", eq2) 
  
# solving the equation 
print("\nValues of 2 unknown variable are as follows:") 
  
print(solve((eq1, eq2), (x, y))) 


display(eq1, eq2)

Equation 1: Eq(x + y, 3)

Equation 2 Eq(x - y, 1)

Values of 2 unknown variable are as follows:
{x: 2, y: 1}

# importing library sympy 
from sympy import symbols, Eq, solve 
  
# defining symbols used in equations 
# or unknown variables 
x, y, z = symbols('x,y,z') 
  
# defining equations 
eq1 = Eq((x+y+z), 1) 
print("\nEquation 1:", eq1) 

  
eq2 = Eq((x-y+2*z), 1) 
print("\nEquation 2", eq2) 

  
eq3 = Eq((2*x-y+2*z), 1) 
print("\nEquation 3", eq3) 

  
# solving the equation and printing the  
# value of unknown variables 
print("\nValues of 3 unknown variable are as follows:") 
print(solve((eq1, eq2, eq3), (x, y, z)))

Equation 1: Eq(x + y + z, 1)

Equation 2 Eq(x - y + 2*z, 1)

Equation 3 Eq(2*x - y + 2*z, 1)

Values of 3 unknown variable are as follows:
{x: 0, y: 1/3, z: 2/3}

#We will solve linear equations using augmented matrix method with Sympy library
#Ref:https://towardsdatascience.com/linear-algebra-systems-of-linear-equations-and-matrices-with-python-d3e0fcb29e85
#Ref:https://www.geeksforgeeks.org/python-sympy-matrix-rref-method/


#The equations are:
# 4x  + 3y = 20
#-5x + 9y = 26

#Let us use the symbolic python library here
import sympy as sp

augmented_matrix = np.array([[4, 3, 20], [-5, 9, 26]])
#augmented_matrix = np.array([[1, 1, 3], [1, -1, 1]])

display(augmented_matrix)

#

#With the help of sympy.Matrix().rref() method, we can put a matrix into reduced Row echelon form.
reduced_echlon_form = sp.Matrix(augmented_matrix).rref()
reduced_echlon_form[0]

array([[ 4,  3, 20],
       [-5,  9, 26]])

#Equations to be solved
# 4x  + 3y = 20
#-5x + 9y = 26

#Start with importing numpy package 
import numpy as np

#The equations are:
# 4x  + 3y = 20
#-5x + 9y = 26

# Define matrices to express above as AxX = B
A = np.array([[4, 3], [-5, 9]])
D = np.array([20, 26])

#Find inverse A
Ainv = np.linalg.inv(A)
print('The inverse of A is: \n', Ainv)

# Transforming the above as X = A-xB  (A- means A inverse)
X = Ainv@D  #@ is the operator for matrix multiplication

print('\n\n The values of the variables are:',X)

The inverse of A is: 
 [[ 0.17647059 -0.05882353]
 [ 0.09803922  0.07843137]]


 The values of the variables are: [2. 4.]

#Let us verify if we have found the correct solution
print("Is the solution correct? Let us verify the value of B and A@X:", D, A@X)

# Now, let us solve below linear equation with three variables 
# 4x + 3y + 2z = 25
# -2x + 2y + 3z = -10
# 3x -5y + 2z = -4

A = np.array([[4, 3, 2], [-2, 2, 3], [3, -5, 2]])
D = np.array([25, -10, -4])
X = np.linalg.inv(A)@D

print(X)

[ 5.  3. -2.]

# Now, let us use np.solve method to solve below linear equation with three variables 
# 4x + 3y + 2z = 25
# -2x + 2y + 3z = -10
# 3x -5y + 2z = -4

A = np.array([[4, 3, 2], [-2, 2, 3], [3, -5, 2]])
D = np.array([25, -10, -4])
X = np.linalg.solve(A,D)

print(X)

[ 5.  3. -2.]

# Import the required libraries 
from scipy import linalg 
import numpy as np 

#Scipy linear algebra: https://www.geeksforgeeks.org/scipy-linear-algebra-scipy-linalg/

  
# The function takes two arrays 
A = np.array([[4, 3, 2], [-2, 2, 3], [3, -5, 2]])
D = np.array([25, -10, -4])
  
# Solving the linear equations 
res = linalg.solve(A, D) 
print(res)

[ 5.  3. -2.]

import numpy as np

#Let us solve the set of linear equation system, where there are dependent equations

# 1  ->  4x + 3y + 2z = 25 
# 2  -> -2x + 2y + 3z = -10
# 3  ->  2x -2y - 3z = 10

#Let us try and solve it

A = np.array([[4, 3, 2], [-2, 2, 3], [2, -2, -3]])
D = np.array([25, -10, 10])
X = np.linalg.solve(A,D)

print(X)

---------------------------------------------------------------------------
LinAlgError                               Traceback (most recent call last)
Cell In[22], line 13
     11 A = np.array([[4, 3, 2], [-2, 2, 3], [2, -2, -3]])
     12 D = np.array([25, -10, 10])
---> 13 X = np.linalg.solve(A,D)
     15 print(X)

File D:\Python_R\Lib\site-packages\numpy\linalg\_linalg.py:413, in solve(a, b)
    410 signature = 'DD->D' if isComplexType(t) else 'dd->d'
    411 with errstate(call=_raise_linalgerror_singular, invalid='call',
    412               over='ignore', divide='ignore', under='ignore'):
--> 413     r = gufunc(a, b, signature=signature)
    415 return wrap(r.astype(result_t, copy=False))

File D:\Python_R\Lib\site-packages\numpy\linalg\_linalg.py:104, in _raise_linalgerror_singular(err, flag)
    103 def _raise_linalgerror_singular(err, flag):
--> 104     raise LinAlgError("Singular matrix")

LinAlgError: Singular matrix

# Since 2 & 3 are dependent equation, let us scrap 3rd equation.
# Now, there are infinite equations possible. To avoid it, let us fix the value of z and find corresponding value of x and y

#So, we write eqution3 as:
# 3  -> z = -2

#Now, let us try and solve it

A = np.array([[4, 3, 2], [-2, 2, 3], [0, 0, 1]])
D = np.array([25, -10, -2])
X = np.linalg.solve(A,D)

print(X)

[ 5.  3. -2.]

01. Week 1: TA Sync Session¶

1.0 Introduction and the agenda¶

1.01 Purpose of the TA sync session¶

1.02 Why Python?¶

1.03 Learning methodology¶

1.04 Agenda for this session¶

1.1 Introduction to Graphing¶

1.1.0 Fundamentals of graphing¶

1.1.1. Graphing an equation using matplotlib¶

1.1.2 Graphing equation using Plotly¶

1.2 Solving linear equations involving multiple variables¶

1.2.0 Types of equation system and its impact on the solution¶

1.2.1 Solving linear equation with two variables with graphing¶

1.2.2 Solving parabolic + linear equation with two variables with graphing¶

Benefits of the Graphing method:¶

Disadvantage of Graphing method:¶

1.2.3 Other methods for solving linear equation system:¶

1.2.4 Solve linear equations using substitution method:¶

1.2.5 Solve linear equations using ECHELON Method¶

1.2.6 Solving equation with two variables using Sympy library¶

1.2.7 Solving equation with multiple variables¶

1.3 Solving linear equations using Matrices¶

1.3.1 Using augmented matrix method¶

1.3.2 Use Numpy to solve linear equation with two variables¶

1.3.3 Use Numpy to solve linear equation with multiple variables¶

1.3.4 Use Numpy .solve method to solve linear equation with multiple variables¶

1.3.5 Use Scipy library to solve linear equation system¶

1.4 Solving Linear equation system with 'dependent' equations¶

Summary and conclusion¶